## Definition

Let $p \in \N$ be a prime number.

Let the $p$-adic valuation on the integers $\nu_p^\Z$ be extended to $\nu_p^\Q: \Q \to \Z \cup \set {+\infty}$ by:

$\map {\nu_p^\Q} {\dfrac a b} := \map {\nu_p^\Z} a - \map {\nu_p^\Z} b$

This mapping $\nu_p^\Q$ is called the $p$-adic valuation (on $\Q$) and is usually denoted $\nu_p: \Q \to \Z \cup \set {+\infty}$.